We study the problem of binomiality of polynomial ideals. Given an ideal with a finite set of generators, we would like to know if there exists a basis for the ideal such that its elements have at most two monomials. Such an ideal is called a binomial ideal. We use the Conradi-Kahle Algorithm for testing whether an ideal is binomial. Our investigations are focused on implementing this algorithm and performing computations. Binomial ideals offer clear computational advantages over arbitrary ideals. They appear in various applications, e.g., in biological and chemical models.
Binomial ideals have been extensively studied in the literature [6, 12, 13]. Eisenbud and Sturmfels in  have shown that Gröbner bases  can be used to test binomiality. Recently, biochemical networks whose steady state ideals are binomial have been studied in the field of Algebraic Systems Biology [5, 7, 18]. Millán and Dickenstein in  have defined MESSI Biological Systems as a general framework for modifications of type enzyme-substrate or swap with intermediates, which includes interesting binomial systems .
In the context of biochemical reaction networks, Millán, Dickenstein, Shiu and Conradi in  present a sufficient condition on the stoichiometric matrix for binomiality of the steady state ideal. Conradi and Kahle  proved that this condition is necessary for homogeneous ideals and proposed an algorithm. The Conradi-Kahle Algorithm is implemented in Macaulay2 . Iosif, Conradi and Kahle in  use the fact that the irreducible components of the varieties of binomial ideals admit monomial parametrization in order to reduce the dimension of detecting total concentrations that lead to multiple steady states.
Our contribution in this article is analysing efficiency and effectiveness of the Conradi-Kahle Algorithm, using Gröbner bases for reduction, applied to some biological models. We first discuss the complexity of the algorithm and reduce it to the complexity of computing a Gröbner basis for a preprocessed input set of polynomials. Then we present our computations in Macaulay2  and Maple  and compare the algorithm with simply computing Gröbner basis of the input ideal which shows the strength of the algorithm. The experiments are performed on biological models in the BioModels repository 111https://www.ebi.ac.uk/biomodels/, which is a repository of mechanistic models of bio-medical systems [2, 8]. Our intial motivation was to understand the advantages and disadvantages of the method in  for testing binomiality of chemical reaction networks. As the Conradi-Kahle Algorithm follows the idea of the method in  with more subtle reduction steps, we rather use the Conradi-Kahle Algorithm to check binomiality of ideals coming from biomodels, although none of our steady state ideals are homogeneous.
2 The Conradi-Kahle Algorithm
The Conradi-Kahle Algorithm is based on the sufficient condition by Millán, Dickenstein, Shiu and Conradi  for binomiality of steady state ideals. The latter states that if the kernel of the stoichiometric matrix has a basis with a particular property then the steady state ideal is binomial. Conradi and Kahle converted this into a sufficient condition for an arbitrary homogenous ideal generated by a set of polynomials of fixed degree. They proved that
is binomial if and only if the reduced row echelon form of the coefficient matrix ofhas at most two non-zero elements in each row. This leads to the Algorithm 1 which is incremental on the degrees of the generators.
Now we analyze the complexity of Algorithm 1.
Steps and . can be ignored.
Step . Let denote the number of distinct monomials in and . Computing the reduced row echelon form of can be done in at most steps, where is the constant in the complexity of matrix multiplication.
Step . needs at most operations which is less or equal than , so we ignore this term.
Steps . can be bounded by , which itself can be bounded by , hence ignored.
Step . This can be done via computing a Gröbner basis of . Another way to do this, is by means of Gaussian elimination on the corresponding Macaulay matrix of .
Step . is equivalent to reducing modulo , which can be done via reducing modulo a Gröbner basis of . Another method to do this is via Gaussian elimination over the Macaulay matrix of .
Following Mayr and Meyer’s work on the complexity of computing Gröbner bases , computations in steps and of the algorithm can be EXP-SPACE. Conradi and Kahle observe through experiments that these steps can be performed via graph enumeration algorithms like breadth first search, which makes it more efficient than Gröbner bases in practice . In this article we do not use such graph enumeration algorithms in our implementations. This is the subject of a future work.
3 Macaulay2 and Maple experiments
We consider 20 Biomodels from the BioModels repository [2, 8] whose steady state ideal is generated by polynomials in where , , are the parameters and are the variables corresponding to the species. Our polynomials are taken from . We use Algorithm 1 to test binomiality of these biomodels. We emphasise that in our computations we do not assign values to the parameters and we work in . We have implemented Algorithm 1 in Maple  and also use a slight variant of the implementation of the algorithm in the Macaulay2 package Binomials [11, 12]. We also test binomiality of an ideal given by a set of generating polynomials via computing a Gröbner basis of the ideal, using Corollary 1.2 in . Our computations are done on a 3.5 GHz Intel Core i7 with 16 GB RAM. In our computations we used Macaulay2 1.12 and Maple 2019.1.
|Biomodel||C-K (M2)||C-K (Maple)||Bin (C-K)||GB (M2)||GB (Maple)||Bin (GB)|
Table 1 shows the results of our computations. Biomodel columns in the table shows the number of the biomodel. The columns C-K (M2) and C-K (Maple) show the CPU timings in seconds of executing Algorithm 1 in Macaulay2 and Maple, respectively. In the column Bin (C-K), Yes means that the algorithm successfully determined that the ideal is binomial, while No means that the algorithm cannot determine whether the ideal is binomial or not. The columns GB (M2) and GB (Maple) are the timings of Gröbner bases computations of the input polynomials in Macaulay2 and Maple, respectively. The Macaulay2 and Maple timings are rounded to the first nonzero digit. Bin (GB) column is blank if the Gröbner basis computation did not finish after 600 seconds. Yes in the latter column means that Gröbner basis computation finished and shows that the ideal is binomial, while No shows that the Gröbner basis computation finished but it detected that the ideal is not binomial.
None of the ideals in the biomodels that we have studied are homogeneous. Therefore, in order to use Algorithm 1 we need to homogenise the ideals. Consequently, if the algorithm returns No, we are not able to say whether the ideal is binomial or not (see [4, Section 4]). As one can see from the column Bin (C-K), the Conradi-Kahle Algorithm is able to test binomiality only for Biomodels , and . If Gröbner bases computations finish, then they can test binomiality for every ideal. However, as one can see from the related columns, this is not the case. Actually in most of the cases, Gröbner bases computations did not finish within 600 seconds. One can see from the table that whenever Gröbner bases computations give a yes answer to the binomiality question, then the Conradi-Kahle Algorithm also can detect this as well. In the Yes cases, the timings for both methods in both Macaulay2 and Maple are very close.
Algorithm 1 returns the output within at most a few
seconds, however, most of the Gröbner bases computations did not
finish in 600 seconds. The advantage of testing binomiality using Gröbner bases computations can be seen in Biomodels , , and ,
where Gröbner bases computations—although slower—show that the ideal is not
binomial, but the Conradi-Kahle Algorithm cannot detect this in spite
of its fast execution. With a few exceptions, we do not observe
significant difference between Macaulay2 and Maple computations,
neither for the Conradi-Kahle Algorithm nor for the Gröbner bases computations. We would like to emphasise that the Conradi-Kahle
Algorithm is complete over homogeneous ideals. However, in this
article we are interested in ideals coming from some biological models
which are inhomogeneous, and this might affect the performance of the
algorithm. In future we will do experiments on homogeneous ideals in
order to better understand the performance of the algorithm in that
Acknowledgement. We would like to thank the anonymous referees for their comments.
-  Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Doctoral dissertation, Mathematical Institute, University of Innsbruck, Innsbruck, Austria (1965)
-  Chelliah, V., Juty, N., Ajmera, I., Ali, R., Dumousseau, M., Glont, M., Hucka, M., Jalowicki, G., Keating, S., Knight-Schrijver, V., Lloret-Villas, A., Nath Natarajan, K., Pettit, J.B., Rodriguez, N., Schubert, M., Wimalaratne, S.M., Zhao, Y., Hermjakob, H., Le Novère, N., Laibe, C.: BioModels: ten-year anniversary. Nucl. Acids Res. (2015)
-  Conradi, C., Iosif, A., Kahle, T.: Multistationarity in the space of total concentrations for systems that admit a monomial parametrization. Bulletin of Mathematical Biology 81(10), 4174–4209 (2019)
-  Conradi, C., Kahle, T.: Detecting binomiality. Advances in Applied Mathematics 71, 52–67 (2015)
-  Craciun, G., Dickenstein, A., Shiu, A., Sturmfels, B.: Toric dynamical systems. Journal of Symbolic Computation 44(11), 1551–1565 (2009)
-  Eisenbud, D., Sturmfels, B., et al.: Binomial ideals. Duke Mathematical Journal 84(1), 1–45 (1996)
-  Gatermann, K., Wolfrum, M.: Bernstein’s second theorem and Viro’s method for sparse polynomial systems in chemistry. Advances in Applied Mathematics 34(2), 252–294 (2005)
-  Glont, M., Nguyen, T.V.N., Graesslin, M., Hälke, R., Ali, R., Schramm, J., Wimalaratne, S.M., Kothamachu, V.B., Rodriguez, N., Swat, M.J., Eils, J., Eils, R., Laibe, C., Malik-Sheriff, R.S., Chelliah, V., Le Novère, N., Hermjakob, H.: Biomodels: expanding horizons to include more modelling approaches and formats. Nucleic Acids Research 46(D1), D1248–D1253 (2018)
-  Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
-  Iosif, A., Rahkooy, H.: MapleBinomials, a Maple package for testing binomiality of ideals, Version v1.0.0 (2019, August 15). Zenodo. Available at http://doi.org/10.5281/zenodo.3564428
-  Kahle, T.: Binomials, a Macaulay 2 package for computations with binomial ideals, Version 1.3, https://github.com/tom111/Binomials/
-  Kahle, T.: Decompositions of binomial ideals. Journal of Software for Algebra and Geometry 4(1), 1–5 (2012)
-  Kahle, T., Rauh, J., Sullivant, S.: Positive margins and primary decomposition. Journal of Commutative Algebra 6(2), 173–208 (2014)
-  Lüders, C., Errami, H., Samal, S.S., Neidhardt, M., Weber, A.: ODEbase: An Extensible Database Providing Algebraic Properties of Dynamical Systems (Extended Abstract). In: CASC 2019: Extended Abstracts. p. 4. Moscow, Russia (2019)
-  Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario: Maple 2019
-  Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. in Math 46, 305–329 (1982)
-  Millán, M.P., Dickenstein, A.: The structure of MESSI biological systems. SIAM Journal on Applied Dynamical Systems 17(2) (2018)
-  Millán, M.P., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bulletin of mathematical biology 74(5), 1027–1065 (2012)